dc.contributor.author Van Vleck, Erik S. dc.contributor.author Mallet-Paret, John dc.contributor.author Cahn, John W. dc.date.accessioned 2015-04-02T16:10:26Z dc.date.available 2015-04-02T16:10:26Z dc.date.issued 1999-03-05 dc.identifier.citation Van Vleck, Erik., Mallet-Paret, John., Cahn, John W. "Traveling Wave Solutions for Systems of ODEs on a Two-Dimensional Spatial Lattice." (1999) SIAM J. Appl. Math., 59(2), 455–493. (39 pages). http://dx.doi.org/10.1137/S0036139996312703. en_US dc.identifier.uri http://hdl.handle.net/1808/17284 dc.description This is the published version, also available here: http://dx.doi.org/10.1137/S0036139996312703. en_US dc.description.abstract We consider infinite systems of ODEs on the two-dimensional integer lattice, given by a bistable scalar ODE at each point, with a nearest neighbor coupling between lattice points. For a class of ideal nonlinearities, we obtain traveling wave solutions in each direction $e^{i\theta}$, and we explore the relation between the wave speed c, the angle $\theta$, and the detuning parameter a of the nonlinearity. Of particular interest is the phenomenon of "propagation failure," and we study how the critical value $a=a^*(\theta)$ depends on $\theta$, where $a^*(\theta)$ is defined as the value of the parameter a at which propagation failure (that is, wave speed c=0) occurs. We show that $a^*:\Bbb{R}\to\Bbb{R} is continuous at each point$\theta$where$\tan\theta$is irrational, and is discontinuous where$\tan\theta\$ is rational or infinite. en_US dc.publisher Society for Industrial and Applied Mathematics en_US dc.subject traveling waves en_US dc.subject propagation failure en_US dc.subject anisotropy en_US dc.subject Allen--Cahn equation en_US dc.title Traveling Wave Solutions for Systems of ODEs on a Two-Dimensional Spatial Lattice en_US dc.type Article kusw.kuauthor Van Vleck, Erik kusw.kudepartment Mathematics en_US dc.identifier.doi 10.1137/S0036139996312703 kusw.oaversion Scholarly/refereed, publisher version kusw.oapolicy This item does not meet KU Open Access policy criteria. dc.rights.accessrights openAccess
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